3.34 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(c g+d g x)^2} \, dx\)

Optimal. Leaf size=102 \[ \frac{A (a+b x)}{g^2 (c+d x) (b c-a d)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g^2 (c+d x) (b c-a d)}-\frac{B n (a+b x)}{g^2 (c+d x) (b c-a d)} \]

[Out]

(A*(a + b*x))/((b*c - a*d)*g^2*(c + d*x)) - (B*n*(a + b*x))/((b*c - a*d)*g^2*(c + d*x)) + (B*(a + b*x)*Log[e*(
(a + b*x)/(c + d*x))^n])/((b*c - a*d)*g^2*(c + d*x))

________________________________________________________________________________________

Rubi [A]  time = 0.0873029, antiderivative size = 107, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{d g^2 (c+d x)}+\frac{b B n \log (a+b x)}{d g^2 (b c-a d)}-\frac{b B n \log (c+d x)}{d g^2 (b c-a d)}+\frac{B n}{d g^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^2,x]

[Out]

(B*n)/(d*g^2*(c + d*x)) + (b*B*n*Log[a + b*x])/(d*(b*c - a*d)*g^2) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(d
*g^2*(c + d*x)) - (b*B*n*Log[c + d*x])/(d*(b*c - a*d)*g^2)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^2} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d g^2 (c+d x)}+\frac{(B n) \int \frac{b c-a d}{g (a+b x) (c+d x)^2} \, dx}{d g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d g^2 (c+d x)}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{d g^2}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d g^2 (c+d x)}+\frac{(B (b c-a d) n) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{d g^2}\\ &=\frac{B n}{d g^2 (c+d x)}+\frac{b B n \log (a+b x)}{d (b c-a d) g^2}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d g^2 (c+d x)}-\frac{b B n \log (c+d x)}{d (b c-a d) g^2}\\ \end{align*}

Mathematica [A]  time = 0.0602251, size = 114, normalized size = 1.12 \[ \frac{B n (b c-a d) \left (\frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2}\right )}{d g^2}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{d g (c g+d g x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(c*g + d*g*x)^2,x]

[Out]

-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(d*g*(c*g + d*g*x))) + (B*(b*c - a*d)*n*(1/((b*c - a*d)*(c + d*x)) +
(b*Log[a + b*x])/(b*c - a*d)^2 - (b*Log[c + d*x])/(b*c - a*d)^2))/(d*g^2)

________________________________________________________________________________________

Maple [F]  time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x)

________________________________________________________________________________________

Maxima [A]  time = 1.17496, size = 184, normalized size = 1.8 \begin{align*} B n{\left (\frac{1}{d^{2} g^{2} x + c d g^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} g^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} g^{2}}\right )} - \frac{B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{d^{2} g^{2} x + c d g^{2}} - \frac{A}{d^{2} g^{2} x + c d g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="maxima")

[Out]

B*n*(1/(d^2*g^2*x + c*d*g^2) + b*log(b*x + a)/((b*c*d - a*d^2)*g^2) - b*log(d*x + c)/((b*c*d - a*d^2)*g^2)) -
B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^2*g^2*x + c*d*g^2) - A/(d^2*g^2*x + c*d*g^2)

________________________________________________________________________________________

Fricas [A]  time = 0.908957, size = 221, normalized size = 2.17 \begin{align*} -\frac{A b c - A a d -{\left (B b c - B a d\right )} n +{\left (B b c - B a d\right )} \log \left (e\right ) -{\left (B b d n x + B a d n\right )} \log \left (\frac{b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} g^{2} x +{\left (b c^{2} d - a c d^{2}\right )} g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="fricas")

[Out]

-(A*b*c - A*a*d - (B*b*c - B*a*d)*n + (B*b*c - B*a*d)*log(e) - (B*b*d*n*x + B*a*d*n)*log((b*x + a)/(d*x + c)))
/((b*c*d^2 - a*d^3)*g^2*x + (b*c^2*d - a*c*d^2)*g^2)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(d*g*x+c*g)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.36895, size = 166, normalized size = 1.63 \begin{align*} \frac{B b n \log \left (b x + a\right )}{b c d g^{2} - a d^{2} g^{2}} - \frac{B b n \log \left (d x + c\right )}{b c d g^{2} - a d^{2} g^{2}} - \frac{B n \log \left (\frac{b x + a}{d x + c}\right )}{d^{2} g^{2} x + c d g^{2}} + \frac{B n - A - B}{d^{2} g^{2} x + c d g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*g*x+c*g)^2,x, algorithm="giac")

[Out]

B*b*n*log(b*x + a)/(b*c*d*g^2 - a*d^2*g^2) - B*b*n*log(d*x + c)/(b*c*d*g^2 - a*d^2*g^2) - B*n*log((b*x + a)/(d
*x + c))/(d^2*g^2*x + c*d*g^2) + (B*n - A - B)/(d^2*g^2*x + c*d*g^2)